(07-13-2019, 01:56 AM)Kernel Sohcahtoa Wrote: I've recently become interested in understanding the theory underlying RSA encryption/decryption. As a result, I learned about RSA via one of my books on number theory. I enjoyed learning about the derivation of the decryption scheme from the encryption scheme, as it made use of Euler's theorem and multiple congruence properties to achieve the desired derivation. There was also a second case of the derivation, which required one to make use of the fact that if a, b, and c are integers with gcd (a,b)=1 and a divides c and b divides c, then ab divides c; I've made use of this fact countless times when doing proofs in number theory.
I actually started my exploration of Cryptography via the Caesar Cipher, since my number theory text started with it. Basically, each letter in the alphabet is converted into an equivalent two digit number. The letter to numbers table goes as follows:
A=00; B=01; C=02; D=03; E=04; F=05; G=06; H=07; I=08
J=09; K=10; L=11; M=12; N=13; O=14; P=15; Q=16; R=17
S=18; T=19; U=20; V=21; W=22; X=23; Y=24; Z=25
We use the Caesar cipher as follows: suppose that we want to encrypt the plain text word "Yogi".
Encryption
1) We convert each letter of the word to its two digit number: Y=25, O=14, G=06, I=08, so our message is expressed numerically as 25 14 06 08 (the spaces are for convenience).
2) We then add 3 to each individual two digit number: 25+3=28, 14+3=17, 06+3=09, 08+3=11, so the encrypted message is now 28 17 09 11. Now, if we obtain a number that is greater than 25, then subtract 26 from it and this will produce the desired encrypted number. Thus, 28 – 26=02. We now substitute 02 for 28 and the encrypted message is 02 17 09 11. Converting this to letters gives the encrypted text CRJL.
Decryption
1) To decrypt the word CRJL, we convert each letter to its two digit number: C=02, R=17, J=09, L=11, so our message is expressed numerically as 02 17 09 11.
2) We then subtract 3 from each individual two digit number: 02 – 3= –1, 17 – 3=14, 09 – 3=06, 11– 3=08, so the decrypted message is–1 14 06 08. Now, if we obtain a number that is less than 0, then add 26 to it and that will produce the desired decrypted number. Thus, –1+26=25. We now substitute 25 for –1 and the decrypted message is 25 14 06 08. Converting this to letters gives YOGI.
For anyone who is interested in practicing, here are some words that can be decrypted using the method described above:
(1) IUHHGRP
Spoiler:
(2) CDQB
Spoiler:
P.S. In math-speak, the actual representation for the encryption scheme is P+3≡C (mod 26) where P is the two digit plain text number and C is the two digit encrypted number. The decryption scheme is represented as C – 3≡P (mod 26).
Hmm. I didn't realize there was going to be math in here today. Just poking at you, Kernel. 
I'm currently reading "The Golden Ratio", by Mario Livio. it's quite interesting. I used to do all kinds of substitution ciphers with friends as a kid. Letters and numbers both.
If you get to thinking you’re a person of some influence, try ordering somebody else’s dog around.
